When linear models are misleading

Jeremy Chiu and Sinem Hacioglu Hoke.

When shocks cause trouble

Small shocks can lead to big crises. At the heart of this issue is that economic dynamics might play out very differently against different backdrops: the same shock would have a very different effect if it hit the economy at the heights of the Great Recession than if it hit during more benign times. It might knock the economy into a more severe and persistent recession or financial stress if it hits already turbulent periods. It seems reasonable, therefore, that we would want to take into account the economic backdrop when we estimate our models.

In a recent staff working paper, we do that by exploring the impact of various shocks in different states of the economy.  Our modelling approach allows us to:

  1. create a non-linear system where it matters what state of the world we are in when the shock hits;
  2. capture low probability but high impact events such as the Great Recession.

Specifically, we build a Threshold VAR (TVAR) model, in a similar spirit of Alessandri and Mumtaz (2014), Galvao and Owyang (2015) and Hubrich and Tetlow (2015), with a simple system summarising the UK real economic activity along with its linkages to (by using UK real GDP growth, inflation and short term interest rates) and the linkages between the banking sector and financial markets (by using UK corporate bond spreads and aggregate bank excess returns).

We consider two variants of the TVAR:

  1. The first tries to distinguish between ‘recessionary’ and ‘non-recessionary’ states of the world by using real GDP growth as the threshold variable. This model is referred as TVAR-Y. TVAR-Y model estimates a threshold level of the real GDP growth above which the economy acts differently than in the state below the threshold.
  2. The second tries to distinguish between ‘financially stressful’ or ‘financially non-stressful’ states of the world by using corporate bond spreads as the threshold variable. We refer to this model as TVAR-S. Similarly, TVAR-S model estimates a threshold level of the corporate bond spreads above which the economy acts differently than in the state below the threshold.

We also consider a linear VAR as a benchmark case.

Different states of the world

We estimate our TVAR models by allowing two states in each. Estimation of the models identifies good and bad states of the UK economy using historical data.

The first specification, TVAR-Y, points us the recessionary and non-recessionary states in Figure 1. The first two recessions coincide with the mid-1970s and were associated with the 1973 oil crisis and subsequent stagflation. The next is the recession in the early 1980s which is often attributed to deflationary government policies including spending cuts and pursuance of monetarism to reduce inflation. The fourth is the early 90s recession which started in the third quarter of 1990 and went on for five quarters. It was primarily associated with high interest rates, falling house prices and an overvalued exchange rate. The last recession was the Great Recession which followed the global financial crisis.

Figure 1: Historical states for TVAR-Y overlapped with UK real GDP growth

Similarly, Figure 2 presents the states in TVAR-S. The first financial stress period corresponds to the second quarter of 1970. The second marks the 1974-5 banking crisis. The third and fourth are associated with 1977:Q1 and 1984:Q2. The former may correspond to the period of introduction of small number of floating-rate issues. The last financial stress periods show the effects of the Great Recession onto the market and overlap with the recessionary states with an exception of the second quarter of 2008.

Figure 2: Historical states for TVAR-S overlapped with UK corporate bond spreadsfigure2

When small shocks cause trouble in bad times

We analyse the dynamic responses of the system when it is hit by various, relatively small, shocks. We particularly explore the effects of adverse output and corporate bond spreads shocks.

When examining what happens when shocks hit during good and bad states, the TVAR models show us that:

  1. Financial shocks worsen recessions. Financial shocks hitting during recessionary periods, identified by the TVAR-Y model, create disproportionately more severe hits to GDP growth than those that hit during non-recessionary periods. (Figure 3). We can infer this by comparing the grey shaded areas, corresponding to the non-recessionary states, and the red shaded areas, corresponding to the recessionary states. The median response of linear BVAR, shown by the green line, falls short of capturing these observations.

Figure 3: Generalised impulse response of real GDP growth to an adverse shock to corporate bond spreads

2. Output shocks worsen financial stress. Figure 4 shows that negative output growth shocks occurring during times of financial stress (red shades), identified by the TVAR-S model, lead to disproportionately higher financial stress than those that hit in non-stressful times (grey shades). When output shocks coincide with financially stressful periods, the resultant increase in corporate bond spreads is seven times as large as when the same shocks hit in the financially non-stressful state.

Figure 4: Generalised impulse response of corporate bond spreads to an adverse shock to real GDP growth


Crucially, the linear BVAR model (green lines) fails to capture these effects in both exercises.  The linear BVAR responses catch, so to say, the (weighted) average response of good and bad states. This highlights the importance of acknowledging nonlinearities.  The TVAR models provide useful information for investigating state dependent responses against structural shocks that the linear BVAR model omits.

When large shocks cause trouble in good times

When economies experience small shocks, as we discussed above, we might not be so worried; they occasionally happen. But sometimes we observe large ones too. What happens then? Brunnermeier and Sannikov (2014) show that large shocks are strongly amplified when the economy is far from its steady state. We also look for empirical evidence for this conclusion.

We aim to explore state specific effects of large shocks. So we conduct impulse response analysis with three-standard-deviation shocks. These are low probability but high impact shocks. If you are in a world where everything is normally distributed, the probability of a similar shock to occur is 0.003 for lower and upper tails combined. Yet they happen as shown by unreported structural shock analysis using our models.

We now focus on the responses to a large corporate bond spread shock in financially stressful and non-stressful periods. For comparison, we first explain what happens when a one-standard-deviation shock hits in different states. Once this small corporate bond spread shock hits the economy, as shown in Figure 5.A, it creates the effect in red in the bad state. Benign time responses are shown in grey. This shock causes a severe and persistent contraction in output growth in the bad state whereas in the good state, output growth recovers rather quickly.

When the large shock hits, as shown in Figure 5.B, the story starts the same but ends differently. The initial responses of corporate spreads and output growth in the bad state (red) are more severe. After the second quarter, the initial shock in spreads starts to die out in the bad state. The response of corporate bond spreads in the good state (grey), however, stays persistently high. This indicates more severe outcomes than the bad state suggests. Eventually, we observe amplified large drop in real output growth in the good state. By the last quarter, the contraction in output growth reaches 1.4 pp – that is thirty times its response to a one-standard-deviation spread shock.

This striking observation is a result of endogenous state switches throughout the forecast horizon. Once a large shock hits, it knocks the good state (financially non-stressful) into the bad state (financially stressful).  The initial response originated in corporate bond spreads then feeds into the real economy. This creates an amplified response and causes output growth to fall further. These findings provide strong empirical support for the theoretical findings of Brunnermeier and Sannikov (2014).

Figure 5: One vs three-standard-deviation shocks to corporate bond spreads and their effect on the real GDP growth (financially stressful state in red and non-stressful state in grey)


The recent Great Recession was a time of extreme events where both financial sector and the real economy suffered heightened stress. Since then researchers and policy makers have paid increasing attention to the potential nonlinearities existing between the two sectors. For instance, policy makers have started developing tools such as regulatory stress testing. Stress testing is designed to simulate the potentially non-linear impact on the economy of shocks being originated from or propagated through the financial sector. Our set-up offers insights on the impact of small and large adverse shocks hitting during certain states of the world. This can be potentially useful in calibrating macro scenarios where multiple adverse shocks hit an economy sequentially.

Jeremy Chiu works in the Bank’s Conjunctural Assessments and Projections Division and Sinem Hacioglu Hoke works in the Bank’s Stress Testing Strategy Division.

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Bank Underground is a blog for Bank of England staff to share views that challenge – or support – prevailing policy orthodoxies. The views expressed here are those of the authors, and are not necessarily those of the Bank of England, or its policy committees.

One thought on “When linear models are misleading

  1. The implication of TVAR-Y recessionary and non-recessionary periods is that after a recessionary period, the economy returns to the non-recessionary state. This is not a realistic assumption. Firstly, after such recessionary periods, the level of GDP is permanently lower than its previous trend path. Secondly, the pattern of sectoral development within the economy after such major fluctuations is quite different from that before the recession.

    Rather than a Normal probability of 0.003 for a three-standard-deviation shock, a more likely probability of 0.114 is obtained with a Breit-Wigner distribution (the ration of normal variables).

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